Thesis Eric

8/3/2019 Thesis Eric

1/169

Further Studies into the Dynamics of a SupercavitatingTorpedo

Eric A. Euteneuer

University of Minnesota

Department of Aerospace Engineering and Mechanics

107 Akerman Hall

110 Union ST SE

Minneapolis, MN 55455

July 18, 2003


2/169

Acknowledgements

I would like to thank Mike Elgersma, Ph.D. for his technical help and guidance on this

work and thesis. His attention to detail and vast dynamics knowledge has saved me an

immense amount of time and headache. I would also like to thank my advisor Prof. Gary

Balas for his guidance and patience throughout this work. Also deserving mention is

Ivan Kirschner at Anteon Corporation. His work, and that of his peers, helped provide

the backbone to this thesis work.

i


3/169

For my Sara Thank you!

ii


4/169

Table of Contents

Further Studies into the Dynamics of a Supercavitating Torpedo....................................... i

Acknowledgements.............................................................................................................. i

Table of Contents............................................................................................................... iiiList of Figures.................................................................................................................... iv

Lift of Tables....................................................................................................................... vList of Symbols.................................................................................................................. vi

Abstract ............................................................................................................................... 1

1 Introduction................................................................................................................. 1

1.1 Focus of Thesis ................................................................................................... 22 General Hydrodynamics ............................................................................................. 3

3 Vehicle Dynamics....................................................................................................... 6

3.1 Coordinate System, States, and Control Variables ............................................. 63.1.1 Flow Angles ................................................................................................ 8

3.1.2 Torpedo Dimensions................................................................................... 83.2 Cavity Dynamics................................................................................................. 93.2.1 Maximum Cavity Dimensions .................................................................. 10

3.2.2 Cavity Centerline ...................................................................................... 10

3.2.3 Cavity Closure .......................................................................................... 15

3.2.4 Final Cavity Shape.................................................................................... 173.3 Cavitator Forces................................................................................................ 18

3.4 Fin Forces.......................................................................................................... 21

3.4.1 Cavity-Fin Interaction............................................................................... 273.4.2 Computation of the Fin Forces and Moments........................................... 28

3.4.3 Notes on Fin Forces .................................................................................. 33

3.5 Planing Forces................................................................................................... 333.5.1 Computation of the Planing Forces........................................................... 37

3.5.1.1 Pressure Forces and Moments .............................................................. 383.5.1.2 Skin Friction Forces.............................................................................. 39

3.5.1.3 Added Mass and Impact Forces............................................................ 40

3.5.1.4 Total Planing Forces and Moments ...................................................... 403.5.2 Effects of Planing...................................................................................... 41

3.6 Mass and Inertial Forces ................................................................................... 42

3.6.1 Center of Mass .......................................................................................... 43

3.6.2 Mass Moments of Inertia .......................................................................... 443.7 Putting it all Together ....................................................................................... 46

3.7.1 EOM about an Arbitrary Point.................................................................. 48

3.7.2 Implementation ......................................................................................... 494 Linearization ............................................................................................................. 51

4.1 General Linearization Procedures and Information.......................................... 52

4.1.1 Limitations of Linearization ..................................................................... 535 Flight Envelope......................................................................................................... 54

6 Stability and System Poles........................................................................................ 56

6.1 Phase Plane Analysis ........................................................................................ 58

iii


5/169

6.2 Simulation and Integration Schemes ................................................................ 58

6.3 Nominal System Stability ................................................................................. 597 Control Law Design.................................................................................................. 63

7.1 Transfer Functions ............................................................................................ 64

7.1.1 1DOF Controller ....................................................................................... 66

7.1.2 Closed-Loop Transfer Functions .............................................................. 677.2 Continuous-Time Linear Quadratic Controller................................................. 68

7.2.1 Closed-Loop Dynamics ............................................................................ 69

7.3 Transformed System......................................................................................... 728 Model Uncertainty .................................................................................................... 74

8.1.1 Open-Loop Effects of Parametric Uncertainty ......................................... 76

8.1.2 General Control Configuration with Uncertainty ..................................... 789 General Conclusions ................................................................................................. 79

10 Bibliography ......................................................................................................... 81

Appendix A: Fin Force and Moment Coefficient Data Computed Directly From AnteonsFin Look-up Table ............................................................................................................ 83

Appendix B ..................................................................................................................... 100Appendix C ..................................................................................................................... 154

Appendix D..................................................................................................................... 156

List of Figures

Figure 1: Schematic of Cavitation Flow Regimes [6] ........................................................ 4

Figure 2: Moment and Angular Rotation Notation............................................................. 7Figure 3: Artists Conception of a Supercavitating Torpedo ............................................... 9

Figure 4: Displacement Model Comparisons ................................................................... 12

Figure 5: Pole Comparison of Systems with Delays vs. "Classic" CenterlineDisplacement............................................................................................................. 15

Figure 6: Cavity Closure Schemes.................................................................................... 16Figure 7 : Cavity Shape Components ............................................................................... 17

Figure 8: Overall Cavity Shape......................................................................................... 17

Figure 9: Cavitator Free-Body Diagram........................................................................... 18

Figure 10: Cavitator Forces and Moments - Test.......................................................... 20Figure 11: Cavitator Forces and Moments - Test ........................................................ 21

Figure 12: Fin Geometry................................................................................................... 22Figure 13: Representation of a Subset of Forces Acting on the Fin and the Appropriate

Flow Regimes ........................................................................................................... 23

Figure 14: Anteon Look-Up Table Data........................................................................... 24Figure 15: Coefficient Data Using Least Squares Approximations.................................. 27

Figure 16: Fin and Supercavity Interaction ...................................................................... 28

Figure 17: Cruciform Orientation of Fins (View from Nose)........................................... 28

Figure 18: Fin Forces and Moments - Test ................................................................... 32Figure 19: Fin Forces and Moments - Test.................................................................... 33Figure 20: Displacement Hull........................................................................................... 34

Figure 21: Planing Hull..................................................................................................... 34

iv


6/169

Figure 22: Possible Supercavitating Flow Schemes with Planing Forces........................ 35

Figure 23: Spring-Mass 2nd Order System with Dead-Zone ........................................... 35Figure 24: Cavity Behavior in an Extreme Turn [1]......................................................... 37

Figure 25: Sketch of Planing Region of Torpedo ............................................................. 38

Figure 26: FFT of Planing Forces..................................................................................... 42

Figure 27: Drawing of Torpedo (In Sectional Form)........................................................ 44Figure 28: Simulation Time Step Comparisons: 1 deg Step in Elevators......................... 50

Figure 29: Torpedo Flight Envelope for Non-Planing Flight ........................................... 55

Figure 30: Nominal System Pole-Zero Map..................................................................... 60Figure 31: Nonlinear Step Responses ............................................................................... 61

Figure 32: Comparison of Nonlinear and Linear System Transients with a 1 deg Elevator

Step ........................................................................................................................... 62Figure 33: Closed-Loop Block Diagrams......................................................................... 66

Figure 34:Closed-Loop Pole-Zero Map of Nominal System with LQR Controller......... 70

Figure 35: Closed-Loop Transients (1 deg/sec Step in r) ................................................. 71Figure 36: Pole-Zero Map of Transformed System.......................................................... 73

Figure 37: System Dynamics Comparison for a Varying Cavitation Number ................. 77Figure 38: Generalized Control Configuration (for Controller Synthesis)....................... 78

Lift of Tables

Table 1: Choices for State Variables .................................................................................. 7

Table 2: Equation Coefficient Values............................................................................... 26

Table 3: Coefficient %-Errors........................................................................................... 26Table 4: Nominal System Pole Information ..................................................................... 59

Table 5: Eigenvectors of Unstable Poles of Nominal System .......................................... 60

Table 6: Uncertain Parameters.......................................................................................... 75

v


7/169

List of Symbols

psv saturated vapor pressure

p free-stream pressure

pc cavity pressure

water density

V free-stream velocity

cavitation number

i cavitation boundary

Fr Froude Number

CQ ventilation coefficient

Q volumetric ventilation flow rate

g gravitational acceleration

Dcav cavitator diameter

Rcav cavitator radius

CDo cavitator drag coefficient at zero angle of attack

x state vector

control vector

angle of attack

side-slip angle

Rc maximum cavity radius

Lc maximum cavity length

rc local cavity radius

c, hc cavity centerline displacements

aturn apparent turn acceleration

ag apparent tail-up accelerationFp perpendicular force acting on the cavitator

lcav moment arm, distance from the cavitator to the origin of the system

cav cavitator apparent angle of attack

cav cavitator apparent side-slip angle

vi


8/169

xcg position of the center of gravity, distance behind the cavitator

cavcav MF , cavitator forces and moments, respectively

imm fin immersion ratio

swp fixed fin sweep

swpf apparent fin sweep

CF, CM fin force and moment coefficients

P vector of coefficients used to calculate the fin force and moment

coefficients

E %-error of the least-squares approximation used for fin coefficient

computations

vector of angles representing the radial locations of the fins

xpiv, rpiv location of the fin pivot points on the torpedo

finfin MF , fin forces and moments, respectively

Lplane length of the torpedo hull that is planing

hplane maximum planing depth

plane planing immersion angle

plane angle measurement of the lateral displacement of the torpedo compared

to the cavity centerline

p radius difference of the cavity and the torpedo at the transom of the

planing section

planeplane MF , planing forces and moments, respectively

J rotational inertia

velocity vector, [u, v, w]T

rotational velocity vector, [p, q, r]T

M mass matrix

rotational velocity matrix

y system output

G system transfer function

d system disturbance

[A,B,C,D] system state-space matrices

vii


9/169

K control law

r reference signal

e error signal

n noise signal

general uncertainty block

viii


10/169

Abstract

Supercavitating torpedoes are complex systems that require an active controller,

which can ensure stability while enabling the torpedo to track a target. In addition, the

control law design process requires a dynamic model that captures the physics of the

problem. It is therefore necessary to define a full 6DOF nonlinear model that lends itself

to linearization for use in the control law design process. This thesis defines such a

model and also discusses such topics as control, model uncertainty, and sensitivity

analysis in order to provide a stepping stone for further studies.

Keywords: Cavitation, Supercavitation, Torpedo, Nonlinearities, Linearization

1 IntroductionAs is known, water is a nearly incompressible medium having properties weakly

changing under great pressure. However, when the pressure in liquid is reduced lower

than the saturated vapor pressure 021.0=svp MPascal, discontinuities in the form of

bubbles, foils and cavities, which are filled by water vapor, are observed in water.

Froude was the first to investigate this phenomenon and gave it the name cavitation,

originating from the Greek word cavity.

The history of hydrodynamics research displays an emphasis on eliminating

cavitation, chiefly because of the erosion, vibration, and acoustical signatures that often

accompany the effect. The drag-reducing benefits of cavitation, however, were noted

during the first half of the last century, and have received significant attention over the

last decade. The invention of the Russian Shkval, a supercavitating torpedo that was

demonstrated in the 1990s, is proof of this. The issue with these torpedoes is that they

currently act like underwater bullets, projectiles that have no active control. In order to

design a control system for these types of vehicles so that they may track targets, the

dynamics must be modeled and analyzed.

There are special conditions that make modeling and control a challenge. The

main difficulties of using the supercavitating flow for underwater objects are connected

1


11/169

with a necessity of ensuring the objects stability in conditions where there is a loss of

Archimedes buoyancy forces and where the location of the center of pressure is well

forward of the center of gravity. Also, whereas a fully-wetted vehicle develops

substantial lift in a turn due to vortex shedding off the hull, a supercavitating vehicle does

not develop significant lift over its gas-enveloped surfaces. These difficulties are in

addition to the highly nonlinear interaction between the cavity and the torpedo body.

However, with proper design, supercavitating vehicles can achieve high velocities

by virtue of reduced drag via a cavitation bubble generated at the nose of the vehicle such

that the skin fraction drag is drastically reduced. Depending on the type and shape of the

supercavitating vehicle under consideration, the overall drag coefficient can be reduced

by an order of magnitude compared to a fully-wetted vehicle.

Currently, the U.S. is pursuing supercavitating marine technology (specifically

torpedoes and other projectiles) and is looking for ways to guaranty stability while

tracking a target through active control, unlike the passively controlled Shkval that only

capable of traveling in straight lines. Supercavitating weapons work in the U.S. is being

directed by the Office of Naval Research (ONR) in Arlington, Va. In general, the ONRs

efforts are aimed at developing two classes of supercavitating technologies: projectiles

and torpedoes. The focus of this thesis is on supercavitating torpedoes.

1.1 Focus of Thesis

The forces on cavitating bodies have been studied at least as far back as the

1920s; an example reference is Brodetsky (1923). Interest increased as focus shifted to

cavitating hydrofoils and propellers; see, for example, Tulin (1958). Since the late 1980s,

the emphasis has returned to nominally axis-symmetric bodies, although vehicle control

requires incorporation of cavitating lifting surface theory as well. Kirschner, et. al.

(1995), Fine and Kinnas (1993), and Savchenko, et al (1997) serve as suitable

introductions and May (1975) is an invaluable resource. The foreign literature contains

several landmark works, for example, Logvinovich, et al, (1985). Given the scope of this

research, the most valuable reference has been Kirschner, et. al at Anteon Corp. [1] This

2


12/169

reference provides the background and basic dynamic model on which most of this

research is based on and provides a model that the ONR is starting to use as a benchmark.

The main point that all these references make is that supercavitating torpedoes are

complex systems, systems that will require the use of an active controller in order to

guarantee stability while performing advanced maneuvers. This controller is necessary to

ensure stability and to enable the torpedo to track a target. However, the control law

design process requires a dynamic model that captures the physics of the problem.

Existing (public) models currently dont model full six degree-of-freedom (6DOF)

dynamics and/or have other issues with them such as mismatching dynamic properties

between the linear and nonlinear models as does the current benchmark model used by

the Office of Naval Research (ONR). This full 6DOF model used by the ONR produces

stable nonlinear transients while the linearized model indicates that the system is

unstable. This prevents the use of the linear representation of the dynamics from being

used in control law design because it does not have the same dynamics as the nonlinear

model and thus eliminates many of the control designers tools.

Therefore it is convenient to define a full 6DOF nonlinear model that lends itself

to linearization for use in the control law design process. This thesis defines such a

model and also discusses such topics as control, model uncertainty, and sensitivity

analysis in order to provide a stepping stone for further studies.

2 General Hydrodynamics

As is known, water is a practically incompressible medium having properties

weakly changing under pressure in hundreds and thousands of atmospheres. However,

when the pressure in the liquid reduces to the saturated vapor pressure, psv = 0.021

MPascals owing to the action of extending stresses, discontinuities on the form of

bubbles, foils and cavities which are filled by water vapor, are observed in water.

Cavitating flows are commonly described by the cavitation number, , and is

expressed as

3


13/169

2

c

2

1V

pp

= Equation 1

where is the fluid density, Vis the free-stream velocity, and and are the ambient

and cavity pressures, respectively. According to the degree, or size of, three cavitation

stages are defined:

p cp

1. Initial cavitation is the bubble stage and it is accompanied by the strong

characteristic noise of collapsing bubbles and is capable of destroying solid

material; for example, blades of screws, pumps, turbines.

2. Partial cavitation is the stage when arising cavities cover a cavitating body part.

The cavity pulses and is unstable.

3. Fully developed cavitation supercavitation is the stage when the cavity

dimensions considerably exceed the body dimensions.

These stages are better illustrated in Figure 1. This figure shows a fictional cavitation

experiment that holds the velocity constant and allows varying amounts of ambient

pressure; various amounts of cavitation can be observed.

Figure 1: Schematic of Cavitation Flow Regimes [6]

4


14/169

Here, i can be thought of as a type of performance boundary where i > results in no

cavitation. For this study, the cavitation number is assumed constant, = 0.029, which is

low enough for natural supercavitation to occur.

Noncavitating flows occur at sufficiently high pressures. Supercavitation occursat very low pressures where a very long vapor cavity exists and in many cases the cavity

wall appears glassy and stable except near the end of the cavity. Limited cavitation is

seen between these two flow regimes.

Other parameters used to describe the supercavitating flows are the Froude (Fr)

number and the ventilation coefficient (CQ) and are shown below (respectively).

cavgD

V=Fr Equation 2

2

cavVD

QCQ = Equation 3

Here g is the gravitational acceleration, the cavitator diameter isDcav, Vis the magnitude

of the vehicles velocity vector, and Q is the volumetric rate at which ventilation gas is

supplied to the cavity. The Froude number characterizes the importance of gravity to the

flow, and therefore governs distortions to the nominally axis-symmetric cavity centerline

shape. The ventilation coefficient governs the time-dependent behavior of the cavity asventilation gas is entrained by the flow. For the trajectories considered in this thesis, the

Froude number is typically on the order of 90 to 110. [1]

A supercavity can be maintained in one of two ways: (1) achieving such a high

speed that the water vaporizes near the nose of the body; or, (2) supplying gas to the

cavity at nearly ambient pressure. The first technique is known as vaporous or natural

cavitation. The second is termed ventilation, or artificial, cavitation. Note that each

concept involves some sort of cavitator with a clean edge to provide the sharp drop in

pressure required to form a clean cavity near the nose of the body. For simplicity, only

natural cavitation is considered in this thesis and thus the ventilation coefficient is zero.

It is, however, conceivable to think of controlling the ventilation, and thus the cavitation

number to affect the dynamics of the system. The effects of varying cavitation numbers

will be described in Section 8.1.1.

5


15/169

3 Vehicle Dynamics

As was mentioned in the introduction, the bulk of this dynamic model is based on,

and expanded from, the work done by Ivan Kirschner et. al. at Anteon Corp under

direction of the ONR. For completeness, all the dynamics will be described in detail

here.

3.1 Coordinate System, States, and Control Variables

The model developed by Anteon uses the cavitator pivot point as the torpedo's

origin. However, for simplicity and for reasons of common convention, this thesis

computes the dynamics with the torpedos center of gravity as the origin.

The coordinate system was chosen to be the same as is defined by Kirschner et.

al. That is, is positive forward of the center of mass, is positive to the starboard

portion of the torpedo, and h is positive, as is defined by the right-hand rule, down. This

coordinate system will make up the body coordinate system. The symbols used

throughout the text correspond generally to current usage and are used in a consistent

manner.The states of the system are the body component states, one set of the two widely

used. The two different choices can be seen in Table 1.

6


16/169

Table 3-1: Choices for State Variables

Body Components Flight Path Components

Variable Symbol Units Variable Symbol Units

Roll Rate p rad/sec Roll Rate p rad/sec

Pitch Rate q rad/sec Pitch Rate q rad/sec

Yaw Rate r rad/sec Yaw Rate r rad/sec

Longitudinal Velocity u ft/sec Velocity Magnitude V ft/sec

Lateral Velocity v ft/sec Sideslip Angle rad

Normal Velocity w ft/sec Angle of Attack rad

Euler Roll Angle rad Bank Angle (about velocity vector) rad

Euler Pitch Angle rad Flight Path Angle rad

Euler Yaw Angle rad Heading Angle rad

North Position ft North Position ft

East Position ft East Position ft

Depth h ft Depth h ft

Appropriate conversions can be seen in [4] and [2] if flight path components of the states

are desired.

The notation defining the positive moments and the positive angular rotations

about the body axes can be seen in Figure 2. Positive velocity components are along the

directions of the axes.

Mz, r

My, q

Mx,p

h

Figure 2: Moment and Angular Rotation Notation

The choice of control surfaces is the same as is presented in Kirchner et. al.

except an additional degree of freedom has been given to the control of the cavitator; in

7


17/169

addition of pivoting in the pitch axis, rotation in the yaw axis has also been considered in

the equations of motion.

Therefore, our states and controls are:

[ ]Trqpwvuhx =

yawpitch cavcavffff

4321=

However, since the water density variation with depth is not modeled in this

investigation, all the position states, North, East, and depth position states, are just

kinematics and play no role in the dynamics modeled below. The state vector then

becomes

[ ]Trqpwvux =

3.1.1 Flow Angles

It is useful to define the orientation of the torpedo about the velocity vector as

these values are often used to compute the forces and moments acting on the body. The

inclination of the body to the velocity vector is defined by the angle of attack and the

sideslip angle such that

u

w1tan = Equation 4

222

1sinwvu

v

++= Equation 5

3.1.2 Torpedo Dimensions

Simulation was conducted on a vehicle with the following characteristics: 4.0 m

in length, 0.2 m in diameter, and with a cavitator diameter of 0.07 m. The fins were

located 3.5 m aft of the cavitator, and were swept back at 45. Although the mass

8


18/169

properties of the vehicle will change as the rocket and ventilation fuels are consumed,

they are assumed constant for purposes of the current analysis. This model is an Applied

Research Lab (ARL) defined model.

3.2 Cavity Dynamics

The behavior of the cavity is central to the dynamics of a supercavitating vehicle.

It is the cavity that makes this dynamical system not only highly nonlinear, but dependent

on the history of the vehicles motion. The nominally steady cavity behavior forms the

basis of the quasi-time dependent model implemented for the current investigation. The

cavity model not only affects the forces acting at the nose of the vehicle, but also has a

strong influence on the fin forces and moments via the amount the fins are immersed in

the free-stream flow and the planing forces, both of which will be discussed in more

detail later in the thesis.

During supercavitation, the cavity stays attached to the body and the cavity

closure is far downstream. The length of the cavity does not vary significantly even

though considerable oscillations can occur at its closure. However, the cavity acts as if it

were an extension of the body. In this case, the same flow field would exist around a

solid body having a shape comprising of the wetted nose plus the free-cavity profile as

might be see in Figure 3.

Figure 3: Artists Conception of a Supercavitating Torpedo

9


19/169

3.2.1 Maximum Cavity Dimensions

The cavity itself is slender, and its maximum diameter is at least 5 times greater

than the cavitator diameter. For axis-symmetric flows, the maximum cavity diameter

(made dimensionless with the cavitator diameter) is a strong function of the cavitator

drag coefficient and the cavitation number, and is otherwise nearly independent of the

cavitator shape (Reichardt, 1946). In fact, both the cavity diameter and the cavity length

increase with cavitator drag and decrease with cavitation number.

Various analytical, numerical, and semi- and fully-empirical models have been

developed that provide estimates of the maximum cavity radius,Rc, and cavity length,Lc.

The analytical formulae of Reichardt (1946) provide useful and reasonably accurate1

approximations for investigation of cavity dynamics. These relations can be seen in

Equations 6-7.

( )2028.01*0

++= DD CC Equation 6

93.035.1 = Dcavc CRR Equation 7

( 6.024.1 123.1 = Dcavc CDL )

Equation 8

where , R.8050.00

constCD == cav is the cavitator radius, and Dcav is the cavitator

diameter.

It is important to note that since the cavitation number is assumed constant for

this investigation, the drag coefficient is considered constant. This means that the

maximum cavity length and radius is assumed constant and will also affect the way that

the cavitator forces are computed. This is a very large assumption since physics dictate

that the cavitation number is going to change as the velocity and cavity change.

3.2.2 Cavity Centerline

There are essentially three methods (which are practical for time-based

simulations) to compute the cavity centerline: (1) Analytical formula developed by

Mnzer and Reichardt (1950) which are described in the paper by Kirschner [1], (2)

1 The approximations are accurate for completely horizontal flows only.

10


20/169

Classic displacement equations based on acceleration, and (3) Use of past (delayed)

position states2

which are coincident with the cavity centerline. Let us first consider

the first two methods which are a function of the instantaneous states.

If the analytical formulas of Mnzer and Reichardt as presented by Kirschner [1]

are used, the local cavity radius, -, and h-offsets for a given distance behind the

cavitator are given in Equations 9-11, respectively.

4.21

2

cavc

cavccav

2/

2//1

=

DL

DLDxRr cc Equation 9

2

cav

turn

2c Fr

1)(

=

D

x

g

ax Equation 10

2

cav

g

2c Fr

1)(

=

D

x

g

axh Equation 11

where Fr is the Froude number, g is gravity, aturn is the apparent turn acceleration and ag

is the apparent tail-up acceleration of the cavity (which are both functions of the states).

ag is also a function of buoyancy, 8.29 m/s2. Although distortions to the cavity shape due

to turning and gravity have been considered, distortions associated with cavitator lift have

been ignored. For more information on how pitching the cavitator can affect cavity

dimensions see reference [9].

If the classic physics equations are used, - and h-offsets are calculated by

Equations 12-13.

( )2

21

=V

xax turnc Equation 12

( )

2

21

= V

x

axh gc Equation 13

Figure 4 shows the drastic difference between the analytical set of equations (Eq.s 10 &

11) derived by Munzer and Reichardt and the classic displacement equations (Eqs 12 &

2 Position states (, , and h) were the option chosen by the researchers at Anteon, though they are not theonly option. Delayed Euler angles would be another suitable option.

11


21/169

13). Notice that the analytical equations (those supposedly derived from experiments)

are much more sensitive to acceleration than the classic physics equations.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Centerline Displacement Comparisons

Position (aft of cavitator) (m)

Displacement(m)

Classic: (1/2)*a*t2

Analytical

Figure 4: Displacement Model Comparisons

In fact, if a velocity of 77 m/s is considered, it turns out that the analytical equations are

nearly 30 times more sensitive to accelerations than classical physics equations predict.

This ultimately shows that if the analytical equations are used, then planing will be much

more likely to occur and thus the model would be much more sensitive to angular

accelerations and turn rates. Since the code produced by Anteon uses the classic physics

equations to project the displacement of the cavity due to buoyancy and because it makes

the model much less sensitive to accelerations, the classic physics equations represented

by Equations 12 and 13 appears to be the better method to model the cavity centerline

displacements in the presence of apparent acceleration.

However, critics will note that assuming that the cavity centerline changes

instantaneously for various accelerations goes against the physics of cavitation and will

argue that the method that uses delayed states should be used. In order to understand the

particulars of the cavity dynamics, consider a projectile moving through a liquid at a

speed that induces natural supercavitation. Now consider a point along the boundary of

the cavity directly behind the cavitator, located at the nose of the projectile. This point is

12


22/169

stationary in the -axis. In other words, the projectile moves, not the boundary. Now this

point is a function of the current states of the projectile. By the time the projectile has

moved and its states have changed, the point along the boundary still is associated with

the original states and is now relatively further behind the cavitator. This delay in states

has an overall effect on the dynamics, but the question is how much?

To answer this, a Dutch-Roll3

model with the slide-slip angle and yaw rate used

as the representative states of the torpedo is considered. After the torpedo dynamics are

identified, two methods are created to compute the cavity centerline; the simple model

uses the classic displacement equations mentioned above, and the second, more complex

model, uses a set of delayed states. The number of delays needed for each state is

dependent upon the number of sections the model designer wants to divide the cavity

profile into; in this case ten sections were chosen and the delays are then assumed to be

variable time delays that are dependent upon the velocity of the torpedo. Interpolation

techniques can then be used to find data between the specified sections. Note that the

more sections that the cavity is divided into, the more accurate the cavity dimensions can

be calculated at any given point behind the cavitator.

In order to take the delays of the more complex system into consideration, it is

common practice to use a nonlinear 2nd

order Pad approximation4

to model this delay.

This not only ensures that the delays are represented in the model, but also guaranteesthat the delays are differentiable. A 2

ndorder approximation is given by the transfer

function written in Equation 14.

( )( ) 1221

12212

2

ss

sse s

++

+ Equation 14

where is the time delay given in seconds.

The consequence of using these approximations is that it adds two poles and

zeros5 to the linear model for every Pad approximation used. This ultimately affects the

3 There are many sources out there that describe how to model the Dutch-Roll dynamics. One of these is

by Etkin [2].4 Other models have tried to model these delays through use of a state buffer in which the states are

stored in an array and then accessed in the functions. This is potentially dangerous as it can result inmisrepresentation of the nonlinear dynamics during linearization.5 Poles, zeros, and other linearization information are described in more depth later in the thesis.

13


23/169

stability and controllability of the linear state-space representation of the nonlinear

system. In order to express how this happens, consider the linearized systemA matrix,

( )

j

i

x

xf

,

, of the Dutch-Roll model based on instantaneous changes in the cavity

centerlinewhich is a function of the torpedo states only and call it A(xt). Now consider a

newA matrix that has components from the Pad approximations and call itA(xPad). The

resultant A matrix is similar to a model based on a nonlinear model which includes the

both the torpedo model and the associated Pad approximations needed for the correct

computation of the cavity shape and would be similar to the one described in Equation

15.

Equation 15

( )(

=

ePad

RollDutch

xAb

axA

A )

Ifa and b are nonzero, it is possible that the system poles would be different from those

just computed from the torpedo states, or the simple model. In addition, in the process

of designing a controller, one would need to design a Kalman filter in order to

approximate the torpedo states before an actual control law could be defined, a possibly

difficult design process in itself.

Now that the differences between the simple and complex systems have been

defined, the dynamics of the two systems can be compared by examining Figure 5.

14


24/169

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0-60

-40

-20

0

20

40

600.84 0.72 0.58 0.44 0.140.3

0.44 0.3 0.14

0.92

0.98

0.92

0.72 0.580.84

80

0.98

204060

Delayed States Method Poles

Delayed States Method Zeros

"Classic" Method Poles

Pole-Zer o Map: Centerline Method Comparison

Real Axis

ImaginaryAxis

Figure 5: Pole Comparison of Systems with Delays vs. "Classic" Centerline Displacement

The dynamics of interest in this graph are shown in the RHP. These poles represent the

dominating dynamics of the two systems. Note that the two poles lie in nearly the same

location. This means that the extra states associated with the 20 delays (ten for each

state, some not shown in the above graph) have little influence on the torpedos motion

(assuming that the steady-state gains are equal). What it also means is that it is

reasonable6

to use the classic displacement equations which are dependent upon the

instantaneous states to compute the distortion of the cavity centerline.

3.2.3 Cavity Closure

A description of the cavity closure zone is the most difficult issue when

describing cavity shape and how it affects the overall cavity dynamics. According to the

6 There is some concern that it will not match in the pitch axis because of the buoyancy forces acting on the

cavity, but the same type of pole matching seen in Figure 5 occurs with a 1-g turn trim condition, acondition similar to straight-and-level flight with buoyancy affecting the cavity shape in the vertical

direction. This leads us to the assumption that the delays will have a similar effect on a full 6DOF.

15


25/169

known theoretical closure schemes (described below) the cavity may be closed on a solid

body (such as a torpedo) in the following manner(s): [5]

Ryabushinsky scheme: A cavity is closed on the solid surface analogous to the

cavitator (Figure 6, a).

Zhukovsky-Roshko scheme: A cavity is closed on the cylinder with diameterDc

equal to the diameter of the biggest cavity section (Figure 6, b).

Brilluene scheme: A cavity is closed on the solid body with a base cavity

formation, where pc2 > po and < 0.7

In this case the base cavity is closed

without a critical point formation (Figure 6, c).

Efros scheme: A cavity is closed with formation of a reentrant jet which may

have effect on the body (Figure 6, d).

Figure 6: Cavity Closure Schemes

The cavity modeled in this study most closely resembles the Ryabushinsky scheme,

similar to the Kutta condition8 with the exception that our model has a more elliptical

shape near the transom rather than the blunt tail depicted in the drawing.

7 Note that this condition states that < 0. THIS IS NOT POSSIBLE! However, this was stated in thesource [6] and I can not translate, or track, the source of this condition in order to correct this apparent typo.8Kutta condition: A body with a sharp trailing edge in motion through a fluid creates about itself a

circulation of sufficient strength to hold the rear stagnation point at the trailing edge of finite angle to make

16


26/169

3.2.4 Final Cavity Shape

When all is said and done, the following ellipsoid represents the cavity shape

(with no other accelerations except buoyancy):

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1Cavity Shape

yc

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.015

-0.01

-0.005

0

zc

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

rc

x

Figure 7 : Cavity Shape Components

Figure 8: Overall Cavity Shape

the flow along the trailing edge bisector angle smooth. For a body with a cusped trailing edge where the

upper and lower surfaces meet tangentially, a smooth flow at the trailing edge requires equal velocities onboth sides of the edge in the tangential direction. Essentially it means that there can be no velocity

discontinuities at the trailing edge, or in this case, the transom (rear) of the cavity.

17


27/169

Remember that a negative h-value means up. If the torpedo were making a starboard

turn, the -component of the cavity centerline would be nonzero and positive and similar

in shape to the h-component.

3.3 Cavitator Forces

Throughout this study of the supercavitating torpedo dynamics, several attempts

were made to describe the disk-cavitator forces. The first attempt didnt proportionally

take into consideration the cavitators relative angle of attack or bank angle and the

second failed to model the lift and side forces correctly. However, by taking the apparent

flow angles into account (as shown in Figure 9), the cavitator forces can be computed as

a function of the disks perpendicular force, apparent angle of attack and apparent

sideslip angle.

cavFp

cav

zcav

ycav

xbody

xcavbod

zbod

Figure 9: Cavitator Free-Body Diagram

where

( ) Dcavp CRVF 2221 = Equation 16

pitchcav

cav

cav

wvu

ql

++=

222Equation 17

yawcav

cav

cav

wvu

rl

++=

222Equation 18

18


28/169

Fp is the perpendicular force acting on the cavitator,and lcav is the distance from the

cavitator to the origin of the system (in this case, the distance to the center of gravity). Fp

is considered perpendicular because the equation used to compute the force is based on

flows that are perpendicular to the cavitator disc.

The body components of the cavitator forces then become:

=

=

cavcavp

cavp

cavcavp

cav

cav

cav

cav

F

F

F

F

F

F

F

z

y

x

cossin

sin

coscos

Equation 19

The moments acting about the cavitators center of effect are assumed to be

negligible, but the moment about the center of gravity due to the forces is not. Since the

origin of the system is the center of gravity and the center of gravity is assumed to lie onthe -axis, the moment arm, lcav, is equal to the location of the center of mass(xcg) and is

measured as the distance aft of the nose of the torpedo. Therefore, the moments

produced by the cavitator forces are:

[ ] cavT

cgcav FxM = 00 Equation 20

Remember that the assumption was made that the drag coefficient remains

constant and so Fp is always constant if the velocity is held constant. This means that the

cavitator forces and moments are only going to be a function of the apparent flow angles

cav and cav .

In order to get a better idea of both the sign convention and sensitivity of the

cavitator forces to the apparent flow angles, a straight-and-level flight condition with a

velocity of 77 m/s is considered and cav and cav are allowed to vary.

19


29/169

-20 -15 -10 -5 0 5 10 15 20-15000

-10000

-5000

0

5000

Fcav

Cavitator Forces - Test

-20 -15 -10 -5 0 5 10 15 20

-1

-0.5

0

0.5

1x 10

4

(deg)

Mcav

Cavitator Moments - Test

Fx

Fy

Fz

Mx

My

Mz

Figure 10: Cavitator Forces and Moments - Test

Notice from Figure 10 that the forces and moments are centered about a negative cav

value. This means that the cavitator has to be pitched down or the angle of attack as to be

negative in order to provide the lift force needed to help support the weight of the torpedo

since fin or planing forces would be insufficient to support the weight alone and since acontrollable forward force is necessary for active control.

Figure 11 shows the effects of varying cav values.

20


30/169

-20 -15 -10 -5 0 5 10 15 20-15000

-10000

-5000

0

5000

Fcav

Cavitator Forces - Test

-20 -15 -10 -5 0 5 10 15 20

-1

-0.5

0

0.5

1x 10

4

(deg)

Mcav

Cavitator Moments - Test

Fx

Fy

Fz

Mx

My

Mz

Figure 11: Cavitator Forces and Moments - Test

Here the cav forces and moments are centered at zero because the torpedo is not in a

turn. If it were, a similar centering shift would be noticed as was noticed in the

example. However, notice that the My value is nonzero for the example. This is

because of the nonzero pitch control of the cavitator.

3.4 Fin Forces

The forces acting on the fins were predicted using a fully three-dimensional

boundary-element method supplemented with a viscous drag correction9. The basic

computational approach is summarized in Fine and Kinnas (1993). A high level of detail

was required over even the limited operational range considered, because several cavity

detachment modes must be taken into account. The simple fin geometry considered for

this investigation (depicted in Figure 12) would be easily fabricated and appropriate to

operation in the supercavitating regime, but is probably not optimal. The wedge shape

9 The source of the viscous drag correction is unknown. It is mentioned by Kirschner in both the paper and

the code, but no direct source was given for this computation so no further explanation can be given.

21


31/169

also provides good strength characteristics. In addition, for small wedge angles, the

unsteady effects associated with the transition between partial cavitation and

supercavitation are confined to a very limited range of deflection angles. [1]

L

S

Wedge Half-AngleFin Geometry

Fin Immersion

Sweepback Angle, Angle of Attack,

[ ] cossinsin 1=

So

Figure 12: Fin Geometry

The forces acting on the cavitating fins are complicated by the different flow

regimes that can be encountered. Some of these regimes include base cavitation, partial

cavitation, and supercavitation.

Body

Cavity

x

y

z

x

y

z

Inflow velocity, V

22


32/169

0.000

0.005

0.010

0.015

0.020

0.025

0.0300.035

0.040

0.045

0.050

0.0 5.0 10.0 15.0

Fz

Angle of Attack (deg)

Partial CavityBase Cavity

Supercavity

x

z

Figure 13: Representation of a Subset of Forces Acting on the Fin and the Appropriate Flow Regimes

The force and moment coefficients were computed with these various cavitating schemes

in mind and were provided via a look-up table computed by Anteon. Each coefficient is

a function of the individual fins apparent angle of attack, apparent sweep, and immersion

ratio, all of which are a function of the local cavity dimensions, fin sweep, and torpedo

velocity (both linear and angular) components.

However, the original fin force and moment coefficient look-up table contained

data that produced a non-differential data space. Recall that one issue with previous

models is that the nonlinear system was not represented in the linearization. One major

contributor of this nonlinear and linear model mismatch actually has to do with the fin

force and moment coefficient computation. The reason behind this inconsistency is

shown graphically in Figures 1-3 in Appendix A which depict the fin forces and moments

for each axis as a function of fin angle of attack and immersion ratio which were

computed directly from the given look-up table for a sweep of 45 degrees. Notice that

there are small symbols on the surface plots. These symbols represent the mapping of

each fin and their associated values during the linearization process. What is important to

notice is that for some forces and moments there are two fins that lie on vertices of the

coefficient data surface. These vertex locations are easy to see in Figure 14 and are not

differentiable. Mathematically, these vertices can be depicted as a type of relay (similar

to an absolute function).

23


33/169

0.50.6

0.7

0.8

0.9

-2-1

0

1

2

-12

-10

-8

-6

-4

-2

0

2

x 10-4

Imm

Fy

for Sweep = 45 deg. (Look-up Table)

Fy

Figure 14: Anteon Look-Up Table Data

Figure 14 clearly illustrates the lines of discontinuity that are present in the y-component

of the fin force coefficient which make the coefficient nondifferentiable. Similar lines of

discontinuity appear in other coefficient values as well. The locations of these lines are

also important. For example, if the system were to be linearized about a straight-and-

level flight condition, the rudder fins (fins 2 and 4) would have zero angle of attack and

thus put their operating space on a line of discontinuity and would make the linearization

invalid. If the linearization were to take place in a different region of the space, (for

example, = 1 deg, imm = 0.75) the linearization may work for a trim condition in this

region, but there is no guarantee that the entire flight envelope of the torpedo would be

differentiable.

This result means that we have to find a new way to represent the fin force and

moment coefficients (or rather smooth the look-up table) so that a linear model may be

24


34/169

computed in order to understand the system properties and to facilitate a (linear) control

law design.

In order to make the fin operating space completely differentiable for all flow

conditions, a parabolic least squares function was fitted to each of the fin force and

moment coefficients. This involves a least-squares type approximation to the fin force

and moment data provided by Anteon such that )fMF swpimmfCC ,,, = . Since the

data is not completely linear, it makes sense to fit a higher order equation to the data. In

indicial notation:

++++= 242

3

3

2

3

1, jijijiji immpimmpimmppC

8765 pimmppimmp jiji +++

Equation 21

Note that the higher the order of the approximation, the more accurate the approximation

will be.

In order to solve for the coefficientspi the system can be solved as follows:

First define the vectors as follows

=

8

8

1

p

p

p

P

=

1

2

2

3

3

j

i

ji

ji

ji

j

i

imm

imm

imm

imm

imm

A

jiCB ,=

Now the coefficientspi can be solved

[ ][ ] BAAAP

BAPAA

BPA

TT

TT

1=

=

=

Equation 22

25


35/169

To get a measure of how good the approximation is to the original data, an averaged

value representing the %-error can be computed using Equation 23.

B

BPAE

= Equation 23

Note that these approximations must be done for each coefficient.

Using this method, the coefficients pi and their representative %-errors can be

seen in Tables 2 & 3.

Coefficient p1 p2 p3 p4

Fx 0.0000E+00 5.0856E-03 4.6230E-05 0.0000E+00Fy 0.0000E+00 -4.8993E-03 -4.4502E-05 0.0000E+00Fz -6.5104E-06 0.0000E+00 0.0000E+00 1.0074E-02

Mx -2.6977E-06 0.0000E+00 0.0000E+00 1.3920E-03My 2.1674E-06 0.0000E+00 0.0000E+00 -1.6503E-03Mz 0.0000E+00 -1.1683E-03 -4.0303E-05 0.0000E+00

Coefficient p5 p6 p7 p8

Fx 0.0000E+00 0.0000E+00 5.6605E-04 -4.7508E-04Fy 0.0000E+00 0.0000E+00 1.0178E-03 1.7334E-04Fz -1.3193E-03 2.6340E-04 0.0000E+00 0.0000E+00Mx 2.1428E-03 -2.8316E-04 0.0000E+00 0.0000E+00My -1.8498E-03 2.7272E-04 0.0000E+00 0.0000E+00Mz 0.0000E+00 0.0000E+00 -1.4744E-03 5.8590E-04

Table 3-2: Equation Coefficient Values

Coefficient % Error

Fx 12.99%Fy 16.71%Fz 11.06%

Mx 14.53%My 13.90%Mz 17.92%

Table 3-3: Coefficient %-Errors

Examples showing the difference between the two coefficient data spaces ((1)

Anteon look-up table data, (2) Least-squares approximations) can be seen in Figures 14

and 15.

26


36/169

0.5

0.6

0.7

0.8

0.9

-2

-1

0

1

2

-15

-10

-5

0

5

x 10-4

Imm

Fy

for Sweep = 45 deg. (Least Squares Fit)

Fy

Figure 15: Coefficient Data Using Least Squares Approximations

Figure 15 shows that by taking a least squares approximate fit to the data, the lines of

discontinuities are removed and the entire operating space of the fins would be

differentiable.

3.4.1 Cavity-Fin Interaction

Note that with the occurrence of cavitation at the fins, there is some interaction

between the supercavity surrounding the entire torpedo and the fin cavities. It has been

shown at the University of Minnesota in water tunnel tests that there will be some loss of

cavity gas (particularly in ventilated cavities) due to the interaction of the fin cavities and

the supercavity enveloping the torpedo.

27


37/169

Figure 16: Fin and Supercavity Interaction

For simplicity, this interaction is ignored. However, it will have an effect on the

cavitation number and thus the dynamics of the model itself. Further studies will need to

explore this interaction and its effects.

3.4.2 Computation of the Fin Forces and MomentsBefore the fin forces and moments are computed, the orientation of the fins have

to be considered. For the model presented here, the fins are arranged in a cruciform

formation as shown in Figure 17.

4

3

1

2

h

Figure 17: Cruciform Orientation of Fins (View from Nose)

This orientation is represented by a vector of angles, ,

where the index is associated with the fin number. Note that this is not the only way to

orient the fins and that the following equations used to compute the fin forces and

moments would work for any fin configuration. Fins 1 and 3 are elevators. Depending

]270180900[oooo=

28


38/169

on the case considered, they provide some component of steady lift to support the

afterbody, and would be important to depth changes. Fins 2 and 4 are rudders that

stabilize the vehicle in roll, and are otherwise deflected only during maneuvers.

In addition to angular placement, the location of the fins on the torpedo body

itself is defined by the variablesxpiv and rpiv where

xpiv = 0.85 Lbody

rpiv = 0.9 Rbody

These positions define the pivot points of the fins.

Also shown in Figure 17 is the sign convention of the fin lift forces (Fz), the

moments (My), and pitch rotation or each fin. The straight arrows on each fin show the

positive direction of the lift force. This type of convention is needed because the fin

force and moment coefficients computed by Anteon were computed for a general wedge-

type fin. This requires special attention to reference frames when computing the total fin

forces in the body reference frame.

The following steps walk through the computations of the individual fin forces and

moments and the appropriate reference frame conversions used to compute the general

fin forces and moments in the body reference frame.

1. Determine the local centerline values for each fin.

( ) ( )( ) ( )( )

( ) ( )( ) (( )iziyiz

iziyiy

cc

cc

finfincen

finfincen

+= )

+=

cossin

sincos

wherecfin

y , , and are the cavity centerline values atxcfin

zcfin

r piv.

2. Find the intersection of the cavity and the local fin.

( ) ( )izriy cenfinR c =2

3. Calculate the fin immersion ratio.

29


39/169

( )( ) ( ))

( )

>


40/169

( ) ( ) ( ) ( ))iswpiiiatk ffinLSCAV cos += Equation 26

9. Compute force and moment coefficients from Anteon precomputed data as a

function ofimm(i), spwf(i), and atk(i).

10.Add any uncertainty associated with the coefficients10.

11.Dimensionalize the force and moment coefficients.

( )

[ ]

[ ] ( )[ ] FixCCCbqM

CCCqF

bwvuq

T

fin

T

MMMfinF

T

FFFF

finF

zyx

zyx

+=

=

++=

00

22222

1

12.Once all the forces and moments are calculated for each fin sum and rotate the

forces and moments into the correct body axis and sum the values

( ) ( )

( ) ( ) ( )( ) ( ) ( )(

( ) ( ) ( )( ) ( ) ( )( )

=

=

=

=

+=

=

fin

fin

fin

N

i

iifin

N

i

iifin

N

i

ifin

iFiFF

iFiFF

FF

1

1

1

cos3sin33

sin3cos22

11

)

))

Equation 27

( ) ( )

( ) ( ) ( )( ) ( ) ((

( ) ( ) ( )( ) ( ) ( )( )

=

=

=

=

+=

=

fin

fin

fin

N

i

iifin

N

i

iifin

N

i

ifin

iMiMM

iMiMM

MM

1

1

1

cos3sin33

sin3cos22

11

Equation 28

Now that the computations of the fin forces and moments have been defined, as

was done for the cavitator forces and moments, the fin forces and moments are shown in

10 System uncertainty is discussed later in the thesis.

31


41/169

Figures 18 and 19 for various flow angles, and . These graphs not only show the

effects of the flow angles on the forces and moments, but also provide insight into the

effects of the fin control surfaces (as they are currently modeled). The effects of various

angles of attack on the fin forces and moments are first shown in Figure 16.

-20 -15 -10 -5 0 5 10 15 20-1

-0.5

0

0.5

1x 10

4

Ffin

Fin Forces - Test

-20 -15 -10 -5 0 5 10 15 20-1

-0.5

0

0.5

1x 104

(deg)

Mfin

Fin Moments - Test

Fx

Fy

Fz

Mx

My

Mz

Figure 18: Fin Forces and Moments - Test

Again, the forces and moments are not centered at zero degrees angle of attack

because the elevator (fins 1 and 3) control values are not set to zero in order to help

support the weight of the torpedo. Notice that unlike the cavitator forces and moments

these forces and moments are nonlinear. This is apparent by the curved lines

representing the dominant fin forces and moments and is the result of different amounts

of cavitation that can occur for various angles of attack. This type of effect was

illustrated in Figures 13-15.

Similar comparisons can be made of the test shown in Figure 19.

32


42/169

-20 -15 -10 -5 0 5 10 15 20-1

-0.5

0

0.5

1x 10

4

Ffin

Fin Forces - Test

-20 -15 -10 -5 0 5 10 15 20

-1

-0.5

0

0.5

1x 10

4

(deg)

Mfin

Fin Moments - Test

Fx

Fy

Fz

Mx

My

Mz

Figure 19: Fin Forces and Moments - Test

3.4.3 Notes on Fin Forces

The fin forces and moment presented here do not specifically model any damping

forces and moments. This ultimately has an effect on the system poles and zeros and is

most important when considering the roll motion. While the sweep of the fins provides a

local lateral force which acts like a passive control surface helping to control pitch and

yaw, there is no such force helping to prevent roll motion. This means that the roll

motion, just based on the fin forces only, is neutrally stable. This makes roll a potentially

hard state to control as the other torpedo dynamics can easily make this motion unstable.

3.5 Planing ForcesCommon planing forces are typically associated with boats. At rest, the planing

hull and displacement hull both displace the water around them.

33


43/169

Figure 20: Displacement Hull

In this case, the planing hull reacts nearly the same as a displacement hull when it

initially gets underway it takes considerable power to produce a small increase in

speed. But at a certain point, external forces acting on the shape cause an interesting

effect, the hull is lifted up onto the surface of the water.

Figure 21: Planing Hull

The planing hull skims along the surface of the water whereas the displacement hull

always forces water around it. This skimming along the waters surface is called

planing. Once on top, the power/speed ratio is considerably altered very little power

is needed to get a large increase in speed. These types of forces occur with

supercavitating torpedo schemes when the torpedo attitude is larger than the allowable

space defined by the cavity shape and dimensions. However, unlike the boat application,

these forces are not desirable for the reason that, while drag may be reduced as compared

to a fully-wetted vehicle, the planing vehicle will produce more drag than the vehicle

entirely enveloped in a supercavity.

Two possible schemes of a planing supercavitating torpedo are shown in Figure

22 (Savchenko et al 1998, Savchenko et al 1999). [5]

34


44/169

Figure 22: Possible Supercavitating Flow Schemes with Planing Forces

In the two possible motion schemes the objects weight G is compensated by two

hydrodynamic forces, 21 YYG += , where Y1 is the lift on the cavitator and Y2 is the lift on

the planing part of the hull. The top part of Figure 22 represents a type of steady-state

planing force that is being used to help support the weight of the body. The bottom part

displays a situation where the body is bouncing around the inside of the cavity.

Upon further investigation into the torpedo models behavior, it is observed that

the planing forces represent a force with a deadzone, much like the one shown in mass-

spring example in Figure 23:

(dampener)

M

(Spring)

Figure 23: Spring-Mass 2nd Order System with Dead-Zone

Here we can see that the spring force will only exist when the end of the spring

hits one of the edges of the mass. The area between the two edges of the mass is the

35


45/169

deadzone. This deadzone is similar to the inside of the cavity and the mass edges are

similar to the cavity boundaries (dimensions).

What this means is that a linearized system would not be representative of all the

possible dynamics. In other words, when planing forces exist, the torpedo is actually a

different system. For this spring mass example shown above, there would have to be

three linear models to represent the three systems: (1) when the spring is not hitting the

edge of the mass, (2) when the spring hits the bottom edge, and (3) when the spring hits

the top edge. A similar process must be applied to the torpedo model for when the

torpedo is planing and when it is not. There are numerous studies in how to handle these

types of nonlinearities in control law design if one decides that it is possible to control the

torpedo (given the very high bandwidth) in the presence of strong and frequent planing

forces.

Planing of a slender afterbody on a supercavitating boundary also distorts the flow

(Logvinovich, 1980). The pressure increase on the wetted portion of the section is

associated with the deflection of the streamlines toward the cavity region. This results in

a jet of fluid into the cavity on each side of the body similar to the spray jet observed

along planing hulls. Both types of secondary flows due to the fins and to the afterbody

planing have been ignored in the current investigation, although the theory used to

estimate the afterbody planing forces accounts for the lowest-order effect of the spray jet.

[1]

Planing forces acting on the blast tube used for propulsion is assumed to be

negligible for reasons that this aft part of the cavity will, in reality, have a large void

fraction and so the hydrodynamic forces acting on the blast tube would be small. Further

studies have been done on the afterbody cavity dynamics by Travis Schauer at the

University of Minnesota and more information regarding these void fractions can be seen

in his Masters thesis.

The importance of cavity distortion in high turn rates is apparent in Figure 24

which represents results for an extreme turn (in this case, a 5-g turn, which is probably

impractical, but is illustrative for the cavity-body interactions important to the dynamics).

36


46/169

Z X

Y

X

Y

Z

Figure 24: Cavity Behavior in an Extreme Turn [1]

The submergence of the afterbody into the flow is clear in this diagram. This is a

cause of nonlinear forces acting on the torpedo. First, as the cavity distorts from a

nominally axisymmetric configuration, the immersion of the fins into the ambient liquid

outside the cavity becomes asymmetric. Therefore, the couple associated with symmetric

or anti-symmetric fin forces and moments will be associated with a nonlinear system

response. Secondly, a supercavitating system designed for a nominally axisymmetric

cavity (or even one designed for cavity tail-up) will be subject to nonlinear forces

associated with afterbody planing.

3.5.1 Computation of the Planing Forces

The planing forces are computed using an extension of Wagner planing theory

developed by Logvinivich (for example, 1980). What this means is that the planing

region of the hull can be approximated as a wedge-type immersion as presented in Figure

25.

37


47/169

Figure 25: Sketch of Planing Region of Torpedo

where

a = Lplane = xentry - xtransom

o = hplane = htransom

= plane = + atan2(hplane,Lplane)

= plane = 0.25(entry+3transom); a conical average weighted to the transom

p = Rcavity at transom Rhull at transom

and correspond to the notation used in Figure 25.

3.5.1.1Pressure Forces and Moments

The drag force associate with planing has two components, pressure drag (form

drag) and skin friction drag (viscous drag). Most of the drag is typically caused by

pressure drag. The pressure drag is caused by a combination of the build of pressure in

front of the submerged portion of the body and the decrease in pressure behind.

The pressure force normal to the inclined longitudinal axis of the cylindrical hullis then given by Equation 29

( )( )

+

+

+=

2

22

cp 12

cossinpplane

p

planeh

planeh

planeplanehhr

hrurF

Equation 29

38


48/169

where rh is the hull radius (assumed to be constant over the planing region), rc, plane, and

are (respectively) the cavity radius (at transom), the angle of attack between the

longitudinal axes of the body and the cavity, and the difference between the cavity and

hull radii (all averaged along the planing region); and is the immersion depth at the

transom measured normal to the cavity centerline.

0h

Similarly, the moment of pressure forces about the transom can be expressed as

pplane

plane

plane

plane

h

h

hr

hrurM

++

+=

2

h

h

plane

222

cp2

cos Equation 30

3.5.1.2Skin Friction Forces

The skin friction forces, caused by the viscosity of water, were computed using

the following set of equations [1]:

7

1

031.0

2

=

=

=

plane

d

planep

h

s

p

plane

c

uLC

hr

u

hu

( )[ ]++

= cccplane

p

hw uuurS arctan1tan

4 2

( )[ ]2212123

1arcsintan2

ssss

planep

h uuuur

+

dwplanef CSuF 22

21 cos= Equation 31

39


49/169

where u is the forward velocity state and the moments are assumed to be negligible.

3.5.1.3Added Mass and Impact Forces

The extra terms that are now added to the planing force computation is the

unsteady force which is proportional to the acceleration and the impact force which is

proportional to the impact speed and the speed of sound in water. The impact force is

important for the case when the hull of the torpedo hits the surface of the cavity.

The generic forces due to acceleration and impact are represented as

Equation 32amCF addedaonaccelerati =

Equation 33pwimpactiimpact VACmCF =

where Cis the speed of sound in water, a and Vare the acceleration and velocities of the

center of mass of the wetted wedge (computed using the norm of the q and rcomponents

of the state-space derivative and state-space, respectively),Apw is the projected area of the

surface area of the wetted wedge, and Ca and Ci are coefficients for the acceleration andimpact forces respectively and are yet to be determined through CFD analysis. Currently,

an upper limit based on a fully wetted cylindrical body, the values ofCa and Ci are 1 and

, respectively. maddedand mimpactare related to the geometry. For a noncavitating sphere

the added mass is equal to half the displaced water, but for a cavitating body, there is no

such compact result. For now, a crude approximation is to set the added mass equal to the

cavity volume and the impact mass to the mass of the displaced water by the impacting

hull.

3.5.1.4Total Planing Forces and Moments

The total planing forces and moments then become

40


50/169

( )( )

++

++=

planeimpactonacceleratip

planeimpactonacceleratip

f

plane

FFF

FFF

F

F

sin

cos Equation 34

+

=

0

0

cos

sin

0 ,wedgecg

plane

cp

cpplane

x

F

M

MM

Equation 35

3.5.2 Effects of Planing

The magnitude of the forces are large (on the order of 6000 N!) and occur at

dominating frequencies11

of about 10 Hz, 25 Hz, and 50 Hz as seen in Figure 26. Keep

in mind that these frequencies and forces are specific to the torpedo geometry described

above as well as the flight condition and may be different for other torpedo models. Due

to the large forces and the high frequency (with dominant modes as high as 50 Hz) of

these forces, not to mention the large increase in drag associated with planing, planing

forces are considered undesirable and are not required for the overall stability of the

torpedo as long as there are other control surfaces such as fins to help support the weight.

In addition, the model becomes a switching model with the cavity dimensions at

the transom of the torpedo representing the dead-zone region. Since the dominant

frequencies of the planing forces are about 10 Hz, 25 Hz, and 50 Hz it would take

significant control effort as well as very fast and expensive actuators and sensors to

actively control the torpedo at this high of a bandwidth. This is why this thesis treats the

planing forces as a general disturbance (which is a function of the states) in the nonlinear

model which means that these forces are not used to compute the linear model used for

control law design. Rather, they will be used to help specify the constraints on the

turning accelerations for the horizontal trajectories considered in order to minimize the

allowable planing forces. In other words, allowable trajectories will be based on the

steady-state turn rate required to produce planing forces.

11 These frequencies are based on the original model developed by Anteon for fin and planing force

supported, straight-and-level flight at a speed of 77 m/s.

41


51/169

0 10 20 30 40 50 60 70 80

0

0.5

1

1.5

2

2.5

3x 10

8

Frequency (Hz)

|F|

FFT of Normal Planing Force

Steady-Flow

All P laning Terms

Figure 26: FFT of Planing Forces

3.6 Mass and Inertial Forces

This thesis treats the torpedo as an ideal rigid body. The term rigidis in reality a

mathematical idealization, because all bodies deform by a certain amount under the

application of loads. If the deformation is small compared to the overall dimensions of

the body, and energy dissipation due to elastic effects is negligible, the rigid body

assumption can safely be used. This is not to say that the high frequency dynamics

associated with an elastic body is not important, but rather that the low frequency

dynamics must be thoroughly understood first.

A rigid body is defined as a body with physical dimensions where the distances

between the particles that constitute the body remain unchanged. One needs to consider

the rotational motion of a rigid body; thus six degrees of freedom, three translational and

three rotational, are required to completely describe the vehicles motion. In addition, one

needs to develop qualities that give information regarding the distribution of mass along

the body. Just as the mass of a body represents its resistance to translational motion, the

42


52/169

distribution of the mass about a certain axis represents the bodys resistance to rotational

motion about that axis.

3.6.1 Center of Mass

A rigid body can be considered as a collection of particles in which the number of

particles approaches infinity and in which the distances between the individual masses

remain the same. AsNapproaches infinity, each particle can be treated as a differential

mass element dm and the mass of the body is computed as an integral over the body

dimensions

= bodydmm

where the nominal mass is set at 175.7 Kg for this investigation.

The location of the center of mass can then be defined as

=body

G dmrm

r1

wherer is the vector from the origin to the differential element dm. Since the torpedo is

symmetric about the -axis, the only nonzero element ofrG is the -component and will

be defined asxcg and is measured in units aft of the cavitator.

The center of mass is a very important quantity, as its use simplifies the analysis

of bodies considerably. One has to perform the integrations above in order to find the

center of mass. These integrals in general are triple integrals, but in order to simplify the

problem the geometry of the torpedo is considered and uniform density is assumed. As is

shown in Figure 27, the torpedo geometry can be broken into four main sections: (1) the

cavitator/pivot joints, (2) nose cone, (3) body cylinder, and (4) the blast tube. Since it is

reasonable to consider most of the mass to be contained in the nose cone and the

cylindrical body, we just need to know the simple geometry of those sections.

43


53/169

xcg

h

Figure 27: Drawing of Torpedo (In Sectional Form)

With just two components of the torpedo represented with simple geometry, it is straight

forward to compute the position of the center of massxcg.

total

cgcylindercgcone

cgm

xmxmx

cylindercone+

= Equation 36

3.6.2 Mass Moments of Inertia

While the center of mass provides valuable information and simplifies the

analysis of translational motion, it gives no measure of the way the mass is distributed on

the body. The mass of a body describes the amount of matter contained in the body and

the resistance of the body to translational motion. The resistance of the body to rotation

is dependant upon how the mass is distributed. This resistance is known as the second

moment of inertia12

or rotational inertia.

A coordinate systemxyz fixed to a point on the body (the center of gravity in this

study) and describe the configuration of a differential mass element by the vectorr = xi +

yj + zk wherex is positive forward of the origin and negative aft of the origin.There are typically two quantities of interest: the distribution of the mass with

respect to a certain axis; and the distribution of mass with respect to a certain plane.

Consider the x-axis first. The perpendicular distance of a differential element dm from

12 The first moment of inertia refers to translational inertia and is just the total mass mtotal.

44


54/169

the x-axis is 22 zyRx += . The mass moment of inertia about the x-axis is then

computed as

dmzydmRJ

bodybody

xxx +== 222 Equation 37

In a similar fashion, the mass moments of inertia about the y and z axes are defined as

Equation 38dmzxdmRJbodybody

yyy +== 222

Equation 39dmyxdmRJbodybody

zzz +== 222

One quick observation is that the mass moment of inertia of a body about a

certain axis becomes larger as the axis is selected further away from the body. This is an

indication that mass moments of inertia will be useful in describing the rotational motion

of a body.

Consider the distribution of the mass with respect to the xy, xz, and yz planes;

these produce the products of inertia.

Equation 40 =body

xy dmxyJ

Equation 41 =body

yzdmyzJ

Equation 42 =body

xz dmxzJ

It is clear thatJxy=Jyx, and so forth. In general, the products of inertia do not contribute

too much to the physical description of the mass distribution, unless there are certain

symmetry properties with respect to the coordinate axes. Since the fins are oriented in

the cross formation and line up with the principal axes, the products of inertia are zero

unless an origin is chosen to be something other than the center of mass.

The moments and products of inertia form the so called inertia matrix, denoted by

[J] and is defined as

45


55/169

Equation 43[ ]

=

zzyzxz

yzyyxy

xzxyxx

JJJ

JJJ

JJJ

J

The diagonal elements of[J] are the principal moments of inertia and they are all positivequantities, each obtained by integration of a positive integrand.

The mass, center of mass, and the inertia matrix of a rigid body specify what are

called the internal properties is the body completely. For an elastic body, one needs to

know measures of the resistance of the body to deformation, in addition to the internal

properties.

To compute the components of the inertia matrix, the simple torpedo geometry

can be taken advantage of again by use of the parallel axis theorem. In other words, the

moments of inertia are computed for the cone and the cylinder separately taking into

consideration the distance from the sections center of gravity to the origin and then

performing the following calculation (parallel axis theorem).

Equation 44

+

+

+

+=22

22

22

yxzyzx

zyzxyx

zxyxzy

iGB

dddddd

dddddd

dddddd

mJJii

where JG are the moments of inertia of the individual section, mi is the mass of thesection, and dx, dy, and dz represent the distances to the origin in the -, -, and h-axes.

The total inertia is then computed by summing the two sections of inertia.

3.7 Putting it all Together

A variety of methods exist for writing equations of motion (EOM) for dynamical

systems. One of the most common is the Newton-Euler formulation. From a historical

perspective, Newton developed his laws for the motion of rigid bodies, even though we

first study them within the context of particles. Defining the inertia force acting on the

body asmacg, Newtons second law can be described as the inertia force being equal and

opposite to the applied forces. The law governing rotational motion was formally stated

by Euler in 1775. The law states that the rate of cha

Documents

Thesis Eric